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Modelling Credit Risk in Portfolios of Consumer Loans: Transition Matrix Model for Consumer Credit Ratings

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Modelling Credit Risk in Portfolios of Consumer Loans: Transition Matrix Model for Consumer Credit Ratings Modelling Credit Risk in portfolios of consumer loans: Transition Matrix Model for Consumer Credit Ratings Madhur Malik and Lyn Thomas 1 School of Management, University of Southampton, United Kingdom, SO17 1BJ Abstract The corporate credit risk literature has many studies modelling the change in the credit risk of corporate bonds over time. There is far less analysis of the credit risk for portfolios of consumer loans. However behavioural scores, which are commonly calculated on a monthly basis by most consumer lenders are the analogues of ratings in corporate credit risk. Motivated by studies in corporate credit risk, we develop a Markov chain model based on behavioural scores to establish the credit risk of portfolios of consumer loans. We motivate the different aspects of the model – the need for a second order Markov chain, the inclusion of economic variables and the age of the loan – using data on a credit card portfolio from a major UK bank. JEL classification: C25; G21; G33 Keywords: Markov chain; Credit risk; Logistic regression; Credit scoring 1. Introduction Since the mid 1980s, banks’ lending to consumers has exceeded that to companies ( Crouhy et al 2001). However it was only with the subprime mortgage crisis of 2007 and the subsequent credit crunch that it was realised what an impact such lending had on the banking sector and also how under researched it is compared with corporate lending models. In particular the need for robust models of the credit risk of portfolios of 1 E-mail: [email protected] 1 consumer loans has been brought into sharp focus by the failure of the ratings agencies to accurately assess the credit risks of Mortgage Backed Securities (MBS) and collateralized debt obligations (CDO) which are based on such portfolios. There are many reasons put forward for the subprime mortgage crisis and the subsequent credit crunch ( Hull 2009, Demyanyk and van Hemert 2008) but one reason that the former led to the latter was the lack of an easily updatable model of the credit risk of portfolios of consumer loans. This lack of suitable models of portfolio level consumer risk was first highlighted during the development of the Basel Accord, when a corporate credit risk model was used to calculate the regulatory capital for all types of loans ( BCBS 2005) even though the basic idea of such a model – that default occurs when debts exceed assets – is not the reason why consumers default. This paper develops a model for the credit risk of portfolios of consumer loans based on behavioural scores for the individual consumers, whose loans make up that portfolio. Such a model would be attractive to lenders, since almost all lenders calculate behavioural scores for all their borrowers on a monthly basis. The behavioural score can be translated into the default probability in a fixed time horizon ( usually one year) in the future for that borrower, but one can consider it as a surrogate for the unobservable creditworthiness of the borrower. We build a Markov chain credit risk model based on behavioural scores for consumers which is related to how the reduced form mark to market corporate credit risk models use a Markov chain approach built on the rating agencies‘ grades, ( Jarrow, Lando, Turnbull 1997). The methodology constructs an empirical forecasting model to derive a multi-period distribution of default rate for long time horizons based on migration matrices built from a historical database of behavioural scores. In our model development we have used the lenders’ behavioural scores but we can use the same methodology on generic bureau scores. This approach helps lenders take long term lending decisions by estimating the risk associated with the change in the quality of portfolio of loans over time. The models also assist in complying with the stress testing requirements in the Basel Accord and other regulations. In addition, the model provides insights on portfolio profitability, the determination of appropriate capital reserves, and creating estimates of portfolio value by generating portfolio level credit loss distributions. 2 There have been a few recent papers which look at modelling the credit risk in consumer loan portfolios. Rosch and Scheule ( Rosch and Scheule 2004) take a variant of the one factor Credit Metrics model , which is the basis of the Basel Accord. They use empirical correlations between different consumer loan types and try to build in economic variables to explain the differences during different parts of the business cycle. Perli and Nayda 2004) also take the corporate credit risk structural models and seek to apply it to consumer lending assuming that a consumer defaults if his assets are lower than a specified threshold. However consumer defaults are usually more about cash flow problems, financial naiveté or fraud and so such a model misses some of the aspects of consumer defaults. Musto and Souleles (2005) use equity pricing as an analogy for changes in the value of consumer loan portfolios, The do look at behavioural scores but take the monthly differences in behavioural scores as the return on assets when applying their equity model. Andrade and Thomas ( 2007) describe a structural model for the credit risk of consumer loans where the behavioural score is a surrogate for the creditworthiness of the borrower. A default occurs if the value of this reputation for creditworthiness , in terms of access to further credit drops below the cost of servicing the debt. Using a case study based on Brazilian credit bureau they found that a random walk was the best model for the idiosyncratic part of creditworthiness. Malik and Thomas (2007) developed a hazard model of time to default for consumer loans where the risk factors were the behavioural score, the age of the loan and economic variables, and used it to develop a credit risk model for portfolios of consumer loans. Bellotti and Crook ( 2008) also used proportional hazards to develop a default risk model for consumer loans. They investigated which economic variables ,might be the most appropriate though they did not use behavioural scores in their model. Thomas (2009b) reviews the consumer credit risk models and points out the analogies with some of the established corporate credit risk models. Since the seminal paper by Jarrow et al ( Jarrow et al 1997), the Markov chain approach has proved popular in modelling the dynamics of the credit risk in corporate portfolios. The idea is to describe the dynamics of the risk in terms of the transition probabilities between the different grades the rating agencies’ award to the firm’s bonds. There are papers which look at how economic conditions as well as the industry sector of the firm 3 affects the transitions matrices, ( Nickell et al 2000) while others generalise the original Jarrow, Lando Turnbull idea, (Hurd and Kuznetsov 2006, ) by using Affine Markov chains or continuous time processes ( Lando Skodeberg 2002). However none of these suggest increasing the order of the Markov chain or considering the age of the loan which are two of the features which we introduce in order to model consumer credit risk using Markov chains. Markov chain models have been used in the consumer lending context before, but none use the behavioural score as the state space nor is the objective of the models to estimate the credit risk at the portfolio level. The first application was by Cyert (1962) who developed a Markov chain model of customer’s repayment behaviour. Subsequently more complex models have been developed by Ho et al (2004) and Trench et al (2003). Schneiderjans and Lock (1994) used Markov chain models to model the marketing aspects of customer relationship management in the banking environment. In section two, we review the properties of behavioural scores and Markov chains, while in section three we describe the Markov chain behavioural score based consumer credit risk model developed. This is parameterised by using cumulative logistic regression to estimate the transition probabilities of the Markov chain. The motivation behind the model and the accuracy of the model’s forecasts are given by means of a case study and section four describes the details of the data used in the case study. Sections five, six and seven give the reasons why one needs to include in the model higher order transition matrices (section five), economic variables to explain the non stationarity of the chain (section six) and the age of the loan (section seven). Section eight describes the full model used, while section nine reports the results of out of time and out of time and out of sample forecasts using the model. The final section draws some conclusions including how the model could be used. It also identifies one issue – which economic variables drive consumer credit risk – where further investigation would benefit all models of consumer credit risk. 2. Behaviour Score Dynamics and Markov Chain models 4 Consumer lenders use behavioural scores updated every month to assess the credit risk of individual borrowers. The score is a sufficient statistic of the probability a borrower will be “Bad” and so default within a certain time horizon (normally taken to be the next twelve months). Borrowers who are not Bad are classified as “Good”. So at time t, a typical borrower with characteristics x(t) ( which may describe recent repayment and usage performance, the current information available at a credit bureau on the borrower, and socio-demographic details) has a score s(x(t),t) so pBxtt( | (),)= pBsxtt ( |((),)) (1) Most scores are log odds score so the direct relationship between the score and the probability of being Bad is given by PG(|((),) sxt t 1 sxtt( ( ), )= log ⇔ PBsxtt ( | ( ( ), )) = s( xt ( ), t ) (2) PBsxtt(|((),) 1 + e Applying Bayes theorem to (2) gives the expansion where if p G(t) is the proportion of the population who are Good at time t (p B(t) is the proportion who are Bad) one has PGsxtt(|((),)pG( t ) PsxttGt (((),)|,) s( x ( t ), t )= log =+ log log =+spop ( t ) woe t ( s ( x ( t ), t )) (3) PBsxtt(|((),) ptB () PsxttBt (((),)|,) The first term is the log of the population odds at time t and the second term is the weight of evidence for that score, (Thomas 2009a).
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